Abstract: Partial subgradient mappings have a key role in the sensitivity analysis of first-order conditions for optimality, and their generalized derivatives are especially important in that respect. It is known that such a mapping is proto-differentiable when it comes from a fully amenable function with compatible parameterization, which is a common case in applications; the proto-derivatives can be evaluated then through projections. Here this result is extended to a still broader class of functions than fully amenable, namely, ones obtained by composing a $C^2$ mapping with a kind of piecewise-$C^2$ convex function under a constraint qualification.